Properties

Label 2128.2.a.k.1.2
Level $2128$
Weight $2$
Character 2128.1
Self dual yes
Analytic conductor $16.992$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2128,2,Mod(1,2128)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2128, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2128.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2128 = 2^{4} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2128.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(16.9921655501\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 532)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.79129\) of defining polynomial
Character \(\chi\) \(=\) 2128.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.79129 q^{3} +3.00000 q^{5} -1.00000 q^{7} +4.79129 q^{9} +O(q^{10})\) \(q+2.79129 q^{3} +3.00000 q^{5} -1.00000 q^{7} +4.79129 q^{9} +3.79129 q^{11} -1.00000 q^{13} +8.37386 q^{15} +3.79129 q^{17} -1.00000 q^{19} -2.79129 q^{21} -4.58258 q^{23} +4.00000 q^{25} +5.00000 q^{27} +3.79129 q^{29} -7.37386 q^{31} +10.5826 q^{33} -3.00000 q^{35} +5.00000 q^{37} -2.79129 q^{39} -3.79129 q^{41} -2.00000 q^{43} +14.3739 q^{45} -10.5826 q^{47} +1.00000 q^{49} +10.5826 q^{51} -8.37386 q^{53} +11.3739 q^{55} -2.79129 q^{57} -12.1652 q^{59} -1.00000 q^{61} -4.79129 q^{63} -3.00000 q^{65} +9.37386 q^{67} -12.7913 q^{69} +12.1652 q^{71} +16.3739 q^{73} +11.1652 q^{75} -3.79129 q^{77} +10.0000 q^{79} -0.417424 q^{81} -14.3739 q^{83} +11.3739 q^{85} +10.5826 q^{87} +7.58258 q^{89} +1.00000 q^{91} -20.5826 q^{93} -3.00000 q^{95} -7.00000 q^{97} +18.1652 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 6 q^{5} - 2 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + 6 q^{5} - 2 q^{7} + 5 q^{9} + 3 q^{11} - 2 q^{13} + 3 q^{15} + 3 q^{17} - 2 q^{19} - q^{21} + 8 q^{25} + 10 q^{27} + 3 q^{29} - q^{31} + 12 q^{33} - 6 q^{35} + 10 q^{37} - q^{39} - 3 q^{41} - 4 q^{43} + 15 q^{45} - 12 q^{47} + 2 q^{49} + 12 q^{51} - 3 q^{53} + 9 q^{55} - q^{57} - 6 q^{59} - 2 q^{61} - 5 q^{63} - 6 q^{65} + 5 q^{67} - 21 q^{69} + 6 q^{71} + 19 q^{73} + 4 q^{75} - 3 q^{77} + 20 q^{79} - 10 q^{81} - 15 q^{83} + 9 q^{85} + 12 q^{87} + 6 q^{89} + 2 q^{91} - 32 q^{93} - 6 q^{95} - 14 q^{97} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.79129 1.61155 0.805775 0.592221i \(-0.201749\pi\)
0.805775 + 0.592221i \(0.201749\pi\)
\(4\) 0 0
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 4.79129 1.59710
\(10\) 0 0
\(11\) 3.79129 1.14312 0.571558 0.820562i \(-0.306339\pi\)
0.571558 + 0.820562i \(0.306339\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) 8.37386 2.16212
\(16\) 0 0
\(17\) 3.79129 0.919522 0.459761 0.888043i \(-0.347935\pi\)
0.459761 + 0.888043i \(0.347935\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −2.79129 −0.609109
\(22\) 0 0
\(23\) −4.58258 −0.955533 −0.477767 0.878487i \(-0.658554\pi\)
−0.477767 + 0.878487i \(0.658554\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) 3.79129 0.704024 0.352012 0.935995i \(-0.385498\pi\)
0.352012 + 0.935995i \(0.385498\pi\)
\(30\) 0 0
\(31\) −7.37386 −1.32438 −0.662192 0.749334i \(-0.730374\pi\)
−0.662192 + 0.749334i \(0.730374\pi\)
\(32\) 0 0
\(33\) 10.5826 1.84219
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) 5.00000 0.821995 0.410997 0.911636i \(-0.365181\pi\)
0.410997 + 0.911636i \(0.365181\pi\)
\(38\) 0 0
\(39\) −2.79129 −0.446964
\(40\) 0 0
\(41\) −3.79129 −0.592100 −0.296050 0.955172i \(-0.595669\pi\)
−0.296050 + 0.955172i \(0.595669\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) 14.3739 2.14273
\(46\) 0 0
\(47\) −10.5826 −1.54363 −0.771814 0.635849i \(-0.780650\pi\)
−0.771814 + 0.635849i \(0.780650\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 10.5826 1.48186
\(52\) 0 0
\(53\) −8.37386 −1.15024 −0.575119 0.818070i \(-0.695044\pi\)
−0.575119 + 0.818070i \(0.695044\pi\)
\(54\) 0 0
\(55\) 11.3739 1.53365
\(56\) 0 0
\(57\) −2.79129 −0.369715
\(58\) 0 0
\(59\) −12.1652 −1.58377 −0.791884 0.610672i \(-0.790900\pi\)
−0.791884 + 0.610672i \(0.790900\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) −4.79129 −0.603646
\(64\) 0 0
\(65\) −3.00000 −0.372104
\(66\) 0 0
\(67\) 9.37386 1.14520 0.572600 0.819835i \(-0.305935\pi\)
0.572600 + 0.819835i \(0.305935\pi\)
\(68\) 0 0
\(69\) −12.7913 −1.53989
\(70\) 0 0
\(71\) 12.1652 1.44374 0.721869 0.692030i \(-0.243283\pi\)
0.721869 + 0.692030i \(0.243283\pi\)
\(72\) 0 0
\(73\) 16.3739 1.91642 0.958208 0.286073i \(-0.0923499\pi\)
0.958208 + 0.286073i \(0.0923499\pi\)
\(74\) 0 0
\(75\) 11.1652 1.28924
\(76\) 0 0
\(77\) −3.79129 −0.432057
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) −0.417424 −0.0463805
\(82\) 0 0
\(83\) −14.3739 −1.57774 −0.788868 0.614562i \(-0.789333\pi\)
−0.788868 + 0.614562i \(0.789333\pi\)
\(84\) 0 0
\(85\) 11.3739 1.23367
\(86\) 0 0
\(87\) 10.5826 1.13457
\(88\) 0 0
\(89\) 7.58258 0.803751 0.401876 0.915694i \(-0.368358\pi\)
0.401876 + 0.915694i \(0.368358\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) −20.5826 −2.13431
\(94\) 0 0
\(95\) −3.00000 −0.307794
\(96\) 0 0
\(97\) −7.00000 −0.710742 −0.355371 0.934725i \(-0.615646\pi\)
−0.355371 + 0.934725i \(0.615646\pi\)
\(98\) 0 0
\(99\) 18.1652 1.82567
\(100\) 0 0
\(101\) −7.74773 −0.770928 −0.385464 0.922723i \(-0.625959\pi\)
−0.385464 + 0.922723i \(0.625959\pi\)
\(102\) 0 0
\(103\) 13.0000 1.28093 0.640464 0.767988i \(-0.278742\pi\)
0.640464 + 0.767988i \(0.278742\pi\)
\(104\) 0 0
\(105\) −8.37386 −0.817205
\(106\) 0 0
\(107\) −0.165151 −0.0159658 −0.00798289 0.999968i \(-0.502541\pi\)
−0.00798289 + 0.999968i \(0.502541\pi\)
\(108\) 0 0
\(109\) −11.7477 −1.12523 −0.562614 0.826720i \(-0.690204\pi\)
−0.562614 + 0.826720i \(0.690204\pi\)
\(110\) 0 0
\(111\) 13.9564 1.32469
\(112\) 0 0
\(113\) 8.37386 0.787747 0.393873 0.919165i \(-0.371135\pi\)
0.393873 + 0.919165i \(0.371135\pi\)
\(114\) 0 0
\(115\) −13.7477 −1.28198
\(116\) 0 0
\(117\) −4.79129 −0.442955
\(118\) 0 0
\(119\) −3.79129 −0.347547
\(120\) 0 0
\(121\) 3.37386 0.306715
\(122\) 0 0
\(123\) −10.5826 −0.954199
\(124\) 0 0
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) −6.74773 −0.598764 −0.299382 0.954133i \(-0.596780\pi\)
−0.299382 + 0.954133i \(0.596780\pi\)
\(128\) 0 0
\(129\) −5.58258 −0.491518
\(130\) 0 0
\(131\) 22.1216 1.93277 0.966386 0.257095i \(-0.0827653\pi\)
0.966386 + 0.257095i \(0.0827653\pi\)
\(132\) 0 0
\(133\) 1.00000 0.0867110
\(134\) 0 0
\(135\) 15.0000 1.29099
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) −9.74773 −0.826791 −0.413396 0.910551i \(-0.635657\pi\)
−0.413396 + 0.910551i \(0.635657\pi\)
\(140\) 0 0
\(141\) −29.5390 −2.48763
\(142\) 0 0
\(143\) −3.79129 −0.317043
\(144\) 0 0
\(145\) 11.3739 0.944548
\(146\) 0 0
\(147\) 2.79129 0.230222
\(148\) 0 0
\(149\) 3.00000 0.245770 0.122885 0.992421i \(-0.460785\pi\)
0.122885 + 0.992421i \(0.460785\pi\)
\(150\) 0 0
\(151\) −13.3739 −1.08835 −0.544175 0.838972i \(-0.683157\pi\)
−0.544175 + 0.838972i \(0.683157\pi\)
\(152\) 0 0
\(153\) 18.1652 1.46857
\(154\) 0 0
\(155\) −22.1216 −1.77685
\(156\) 0 0
\(157\) −20.1216 −1.60588 −0.802939 0.596061i \(-0.796731\pi\)
−0.802939 + 0.596061i \(0.796731\pi\)
\(158\) 0 0
\(159\) −23.3739 −1.85367
\(160\) 0 0
\(161\) 4.58258 0.361158
\(162\) 0 0
\(163\) −4.37386 −0.342587 −0.171294 0.985220i \(-0.554795\pi\)
−0.171294 + 0.985220i \(0.554795\pi\)
\(164\) 0 0
\(165\) 31.7477 2.47156
\(166\) 0 0
\(167\) −13.4174 −1.03827 −0.519136 0.854692i \(-0.673746\pi\)
−0.519136 + 0.854692i \(0.673746\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −4.79129 −0.366399
\(172\) 0 0
\(173\) 22.7477 1.72948 0.864739 0.502222i \(-0.167484\pi\)
0.864739 + 0.502222i \(0.167484\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) 0 0
\(177\) −33.9564 −2.55232
\(178\) 0 0
\(179\) 15.7913 1.18030 0.590148 0.807295i \(-0.299069\pi\)
0.590148 + 0.807295i \(0.299069\pi\)
\(180\) 0 0
\(181\) −9.37386 −0.696754 −0.348377 0.937355i \(-0.613267\pi\)
−0.348377 + 0.937355i \(0.613267\pi\)
\(182\) 0 0
\(183\) −2.79129 −0.206338
\(184\) 0 0
\(185\) 15.0000 1.10282
\(186\) 0 0
\(187\) 14.3739 1.05112
\(188\) 0 0
\(189\) −5.00000 −0.363696
\(190\) 0 0
\(191\) −23.5390 −1.70322 −0.851612 0.524173i \(-0.824374\pi\)
−0.851612 + 0.524173i \(0.824374\pi\)
\(192\) 0 0
\(193\) 7.37386 0.530782 0.265391 0.964141i \(-0.414499\pi\)
0.265391 + 0.964141i \(0.414499\pi\)
\(194\) 0 0
\(195\) −8.37386 −0.599665
\(196\) 0 0
\(197\) −2.37386 −0.169131 −0.0845654 0.996418i \(-0.526950\pi\)
−0.0845654 + 0.996418i \(0.526950\pi\)
\(198\) 0 0
\(199\) 23.7477 1.68343 0.841716 0.539921i \(-0.181546\pi\)
0.841716 + 0.539921i \(0.181546\pi\)
\(200\) 0 0
\(201\) 26.1652 1.84555
\(202\) 0 0
\(203\) −3.79129 −0.266096
\(204\) 0 0
\(205\) −11.3739 −0.794385
\(206\) 0 0
\(207\) −21.9564 −1.52608
\(208\) 0 0
\(209\) −3.79129 −0.262249
\(210\) 0 0
\(211\) 24.3739 1.67797 0.838983 0.544158i \(-0.183151\pi\)
0.838983 + 0.544158i \(0.183151\pi\)
\(212\) 0 0
\(213\) 33.9564 2.32666
\(214\) 0 0
\(215\) −6.00000 −0.409197
\(216\) 0 0
\(217\) 7.37386 0.500570
\(218\) 0 0
\(219\) 45.7042 3.08840
\(220\) 0 0
\(221\) −3.79129 −0.255030
\(222\) 0 0
\(223\) 23.7477 1.59027 0.795133 0.606435i \(-0.207401\pi\)
0.795133 + 0.606435i \(0.207401\pi\)
\(224\) 0 0
\(225\) 19.1652 1.27768
\(226\) 0 0
\(227\) 3.79129 0.251637 0.125818 0.992053i \(-0.459844\pi\)
0.125818 + 0.992053i \(0.459844\pi\)
\(228\) 0 0
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) −10.5826 −0.696282
\(232\) 0 0
\(233\) −28.1216 −1.84231 −0.921153 0.389200i \(-0.872752\pi\)
−0.921153 + 0.389200i \(0.872752\pi\)
\(234\) 0 0
\(235\) −31.7477 −2.07099
\(236\) 0 0
\(237\) 27.9129 1.81314
\(238\) 0 0
\(239\) 7.74773 0.501159 0.250579 0.968096i \(-0.419379\pi\)
0.250579 + 0.968096i \(0.419379\pi\)
\(240\) 0 0
\(241\) −8.74773 −0.563491 −0.281745 0.959489i \(-0.590913\pi\)
−0.281745 + 0.959489i \(0.590913\pi\)
\(242\) 0 0
\(243\) −16.1652 −1.03699
\(244\) 0 0
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) 1.00000 0.0636285
\(248\) 0 0
\(249\) −40.1216 −2.54260
\(250\) 0 0
\(251\) −9.79129 −0.618021 −0.309010 0.951059i \(-0.599998\pi\)
−0.309010 + 0.951059i \(0.599998\pi\)
\(252\) 0 0
\(253\) −17.3739 −1.09229
\(254\) 0 0
\(255\) 31.7477 1.98812
\(256\) 0 0
\(257\) 11.2087 0.699180 0.349590 0.936903i \(-0.386321\pi\)
0.349590 + 0.936903i \(0.386321\pi\)
\(258\) 0 0
\(259\) −5.00000 −0.310685
\(260\) 0 0
\(261\) 18.1652 1.12439
\(262\) 0 0
\(263\) −6.79129 −0.418769 −0.209384 0.977833i \(-0.567146\pi\)
−0.209384 + 0.977833i \(0.567146\pi\)
\(264\) 0 0
\(265\) −25.1216 −1.54321
\(266\) 0 0
\(267\) 21.1652 1.29529
\(268\) 0 0
\(269\) −9.95644 −0.607055 −0.303527 0.952823i \(-0.598164\pi\)
−0.303527 + 0.952823i \(0.598164\pi\)
\(270\) 0 0
\(271\) 14.1216 0.857826 0.428913 0.903346i \(-0.358897\pi\)
0.428913 + 0.903346i \(0.358897\pi\)
\(272\) 0 0
\(273\) 2.79129 0.168936
\(274\) 0 0
\(275\) 15.1652 0.914493
\(276\) 0 0
\(277\) 3.25227 0.195410 0.0977051 0.995215i \(-0.468850\pi\)
0.0977051 + 0.995215i \(0.468850\pi\)
\(278\) 0 0
\(279\) −35.3303 −2.11517
\(280\) 0 0
\(281\) 19.4174 1.15835 0.579173 0.815205i \(-0.303375\pi\)
0.579173 + 0.815205i \(0.303375\pi\)
\(282\) 0 0
\(283\) 6.37386 0.378887 0.189443 0.981892i \(-0.439332\pi\)
0.189443 + 0.981892i \(0.439332\pi\)
\(284\) 0 0
\(285\) −8.37386 −0.496025
\(286\) 0 0
\(287\) 3.79129 0.223793
\(288\) 0 0
\(289\) −2.62614 −0.154479
\(290\) 0 0
\(291\) −19.5390 −1.14540
\(292\) 0 0
\(293\) 19.7477 1.15367 0.576837 0.816859i \(-0.304287\pi\)
0.576837 + 0.816859i \(0.304287\pi\)
\(294\) 0 0
\(295\) −36.4955 −2.12485
\(296\) 0 0
\(297\) 18.9564 1.09996
\(298\) 0 0
\(299\) 4.58258 0.265017
\(300\) 0 0
\(301\) 2.00000 0.115278
\(302\) 0 0
\(303\) −21.6261 −1.24239
\(304\) 0 0
\(305\) −3.00000 −0.171780
\(306\) 0 0
\(307\) 27.3739 1.56231 0.781154 0.624338i \(-0.214631\pi\)
0.781154 + 0.624338i \(0.214631\pi\)
\(308\) 0 0
\(309\) 36.2867 2.06428
\(310\) 0 0
\(311\) −26.2087 −1.48616 −0.743080 0.669203i \(-0.766636\pi\)
−0.743080 + 0.669203i \(0.766636\pi\)
\(312\) 0 0
\(313\) 12.7477 0.720544 0.360272 0.932847i \(-0.382684\pi\)
0.360272 + 0.932847i \(0.382684\pi\)
\(314\) 0 0
\(315\) −14.3739 −0.809875
\(316\) 0 0
\(317\) 14.8348 0.833208 0.416604 0.909088i \(-0.363220\pi\)
0.416604 + 0.909088i \(0.363220\pi\)
\(318\) 0 0
\(319\) 14.3739 0.804782
\(320\) 0 0
\(321\) −0.460985 −0.0257297
\(322\) 0 0
\(323\) −3.79129 −0.210953
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) 0 0
\(327\) −32.7913 −1.81336
\(328\) 0 0
\(329\) 10.5826 0.583436
\(330\) 0 0
\(331\) −28.3739 −1.55957 −0.779784 0.626048i \(-0.784671\pi\)
−0.779784 + 0.626048i \(0.784671\pi\)
\(332\) 0 0
\(333\) 23.9564 1.31280
\(334\) 0 0
\(335\) 28.1216 1.53645
\(336\) 0 0
\(337\) 1.37386 0.0748391 0.0374196 0.999300i \(-0.488086\pi\)
0.0374196 + 0.999300i \(0.488086\pi\)
\(338\) 0 0
\(339\) 23.3739 1.26949
\(340\) 0 0
\(341\) −27.9564 −1.51393
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −38.3739 −2.06598
\(346\) 0 0
\(347\) −0.791288 −0.0424786 −0.0212393 0.999774i \(-0.506761\pi\)
−0.0212393 + 0.999774i \(0.506761\pi\)
\(348\) 0 0
\(349\) −1.62614 −0.0870451 −0.0435225 0.999052i \(-0.513858\pi\)
−0.0435225 + 0.999052i \(0.513858\pi\)
\(350\) 0 0
\(351\) −5.00000 −0.266880
\(352\) 0 0
\(353\) −8.53901 −0.454486 −0.227243 0.973838i \(-0.572971\pi\)
−0.227243 + 0.973838i \(0.572971\pi\)
\(354\) 0 0
\(355\) 36.4955 1.93698
\(356\) 0 0
\(357\) −10.5826 −0.560089
\(358\) 0 0
\(359\) 0.626136 0.0330462 0.0165231 0.999863i \(-0.494740\pi\)
0.0165231 + 0.999863i \(0.494740\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 9.41742 0.494287
\(364\) 0 0
\(365\) 49.1216 2.57114
\(366\) 0 0
\(367\) 22.4955 1.17425 0.587127 0.809495i \(-0.300259\pi\)
0.587127 + 0.809495i \(0.300259\pi\)
\(368\) 0 0
\(369\) −18.1652 −0.945640
\(370\) 0 0
\(371\) 8.37386 0.434749
\(372\) 0 0
\(373\) 16.3739 0.847807 0.423903 0.905707i \(-0.360660\pi\)
0.423903 + 0.905707i \(0.360660\pi\)
\(374\) 0 0
\(375\) −8.37386 −0.432424
\(376\) 0 0
\(377\) −3.79129 −0.195261
\(378\) 0 0
\(379\) −21.7477 −1.11711 −0.558553 0.829469i \(-0.688643\pi\)
−0.558553 + 0.829469i \(0.688643\pi\)
\(380\) 0 0
\(381\) −18.8348 −0.964939
\(382\) 0 0
\(383\) −21.0000 −1.07305 −0.536525 0.843884i \(-0.680263\pi\)
−0.536525 + 0.843884i \(0.680263\pi\)
\(384\) 0 0
\(385\) −11.3739 −0.579666
\(386\) 0 0
\(387\) −9.58258 −0.487110
\(388\) 0 0
\(389\) −18.6261 −0.944383 −0.472191 0.881496i \(-0.656537\pi\)
−0.472191 + 0.881496i \(0.656537\pi\)
\(390\) 0 0
\(391\) −17.3739 −0.878634
\(392\) 0 0
\(393\) 61.7477 3.11476
\(394\) 0 0
\(395\) 30.0000 1.50946
\(396\) 0 0
\(397\) −5.25227 −0.263604 −0.131802 0.991276i \(-0.542076\pi\)
−0.131802 + 0.991276i \(0.542076\pi\)
\(398\) 0 0
\(399\) 2.79129 0.139739
\(400\) 0 0
\(401\) −12.6261 −0.630519 −0.315260 0.949005i \(-0.602092\pi\)
−0.315260 + 0.949005i \(0.602092\pi\)
\(402\) 0 0
\(403\) 7.37386 0.367318
\(404\) 0 0
\(405\) −1.25227 −0.0622259
\(406\) 0 0
\(407\) 18.9564 0.939636
\(408\) 0 0
\(409\) 27.1216 1.34108 0.670538 0.741875i \(-0.266063\pi\)
0.670538 + 0.741875i \(0.266063\pi\)
\(410\) 0 0
\(411\) 16.7477 0.826104
\(412\) 0 0
\(413\) 12.1652 0.598608
\(414\) 0 0
\(415\) −43.1216 −2.11676
\(416\) 0 0
\(417\) −27.2087 −1.33242
\(418\) 0 0
\(419\) 25.7477 1.25786 0.628929 0.777462i \(-0.283493\pi\)
0.628929 + 0.777462i \(0.283493\pi\)
\(420\) 0 0
\(421\) −19.0000 −0.926003 −0.463002 0.886357i \(-0.653228\pi\)
−0.463002 + 0.886357i \(0.653228\pi\)
\(422\) 0 0
\(423\) −50.7042 −2.46532
\(424\) 0 0
\(425\) 15.1652 0.735618
\(426\) 0 0
\(427\) 1.00000 0.0483934
\(428\) 0 0
\(429\) −10.5826 −0.510932
\(430\) 0 0
\(431\) 6.00000 0.289010 0.144505 0.989504i \(-0.453841\pi\)
0.144505 + 0.989504i \(0.453841\pi\)
\(432\) 0 0
\(433\) 9.74773 0.468446 0.234223 0.972183i \(-0.424745\pi\)
0.234223 + 0.972183i \(0.424745\pi\)
\(434\) 0 0
\(435\) 31.7477 1.52219
\(436\) 0 0
\(437\) 4.58258 0.219214
\(438\) 0 0
\(439\) −41.4955 −1.98047 −0.990235 0.139408i \(-0.955480\pi\)
−0.990235 + 0.139408i \(0.955480\pi\)
\(440\) 0 0
\(441\) 4.79129 0.228157
\(442\) 0 0
\(443\) 14.0436 0.667230 0.333615 0.942709i \(-0.391732\pi\)
0.333615 + 0.942709i \(0.391732\pi\)
\(444\) 0 0
\(445\) 22.7477 1.07835
\(446\) 0 0
\(447\) 8.37386 0.396070
\(448\) 0 0
\(449\) 25.1216 1.18556 0.592781 0.805364i \(-0.298030\pi\)
0.592781 + 0.805364i \(0.298030\pi\)
\(450\) 0 0
\(451\) −14.3739 −0.676839
\(452\) 0 0
\(453\) −37.3303 −1.75393
\(454\) 0 0
\(455\) 3.00000 0.140642
\(456\) 0 0
\(457\) −39.8693 −1.86501 −0.932504 0.361160i \(-0.882381\pi\)
−0.932504 + 0.361160i \(0.882381\pi\)
\(458\) 0 0
\(459\) 18.9564 0.884811
\(460\) 0 0
\(461\) −11.5390 −0.537426 −0.268713 0.963220i \(-0.586598\pi\)
−0.268713 + 0.963220i \(0.586598\pi\)
\(462\) 0 0
\(463\) 34.4955 1.60314 0.801570 0.597901i \(-0.203998\pi\)
0.801570 + 0.597901i \(0.203998\pi\)
\(464\) 0 0
\(465\) −61.7477 −2.86348
\(466\) 0 0
\(467\) −34.2867 −1.58660 −0.793301 0.608830i \(-0.791639\pi\)
−0.793301 + 0.608830i \(0.791639\pi\)
\(468\) 0 0
\(469\) −9.37386 −0.432845
\(470\) 0 0
\(471\) −56.1652 −2.58795
\(472\) 0 0
\(473\) −7.58258 −0.348647
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) −40.1216 −1.83704
\(478\) 0 0
\(479\) −12.6261 −0.576903 −0.288451 0.957495i \(-0.593140\pi\)
−0.288451 + 0.957495i \(0.593140\pi\)
\(480\) 0 0
\(481\) −5.00000 −0.227980
\(482\) 0 0
\(483\) 12.7913 0.582024
\(484\) 0 0
\(485\) −21.0000 −0.953561
\(486\) 0 0
\(487\) −11.0000 −0.498458 −0.249229 0.968445i \(-0.580177\pi\)
−0.249229 + 0.968445i \(0.580177\pi\)
\(488\) 0 0
\(489\) −12.2087 −0.552097
\(490\) 0 0
\(491\) 15.1652 0.684394 0.342197 0.939628i \(-0.388829\pi\)
0.342197 + 0.939628i \(0.388829\pi\)
\(492\) 0 0
\(493\) 14.3739 0.647366
\(494\) 0 0
\(495\) 54.4955 2.44939
\(496\) 0 0
\(497\) −12.1652 −0.545682
\(498\) 0 0
\(499\) −17.6261 −0.789054 −0.394527 0.918884i \(-0.629092\pi\)
−0.394527 + 0.918884i \(0.629092\pi\)
\(500\) 0 0
\(501\) −37.4519 −1.67323
\(502\) 0 0
\(503\) 6.33030 0.282254 0.141127 0.989991i \(-0.454927\pi\)
0.141127 + 0.989991i \(0.454927\pi\)
\(504\) 0 0
\(505\) −23.2432 −1.03431
\(506\) 0 0
\(507\) −33.4955 −1.48759
\(508\) 0 0
\(509\) 0.330303 0.0146404 0.00732021 0.999973i \(-0.497670\pi\)
0.00732021 + 0.999973i \(0.497670\pi\)
\(510\) 0 0
\(511\) −16.3739 −0.724337
\(512\) 0 0
\(513\) −5.00000 −0.220755
\(514\) 0 0
\(515\) 39.0000 1.71855
\(516\) 0 0
\(517\) −40.1216 −1.76455
\(518\) 0 0
\(519\) 63.4955 2.78714
\(520\) 0 0
\(521\) −13.4174 −0.587828 −0.293914 0.955832i \(-0.594958\pi\)
−0.293914 + 0.955832i \(0.594958\pi\)
\(522\) 0 0
\(523\) −0.252273 −0.0110311 −0.00551556 0.999985i \(-0.501756\pi\)
−0.00551556 + 0.999985i \(0.501756\pi\)
\(524\) 0 0
\(525\) −11.1652 −0.487287
\(526\) 0 0
\(527\) −27.9564 −1.21780
\(528\) 0 0
\(529\) −2.00000 −0.0869565
\(530\) 0 0
\(531\) −58.2867 −2.52943
\(532\) 0 0
\(533\) 3.79129 0.164219
\(534\) 0 0
\(535\) −0.495454 −0.0214204
\(536\) 0 0
\(537\) 44.0780 1.90211
\(538\) 0 0
\(539\) 3.79129 0.163302
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 0 0
\(543\) −26.1652 −1.12285
\(544\) 0 0
\(545\) −35.2432 −1.50965
\(546\) 0 0
\(547\) 12.3739 0.529068 0.264534 0.964376i \(-0.414782\pi\)
0.264534 + 0.964376i \(0.414782\pi\)
\(548\) 0 0
\(549\) −4.79129 −0.204487
\(550\) 0 0
\(551\) −3.79129 −0.161514
\(552\) 0 0
\(553\) −10.0000 −0.425243
\(554\) 0 0
\(555\) 41.8693 1.77725
\(556\) 0 0
\(557\) 0.626136 0.0265303 0.0132651 0.999912i \(-0.495777\pi\)
0.0132651 + 0.999912i \(0.495777\pi\)
\(558\) 0 0
\(559\) 2.00000 0.0845910
\(560\) 0 0
\(561\) 40.1216 1.69393
\(562\) 0 0
\(563\) 12.1652 0.512700 0.256350 0.966584i \(-0.417480\pi\)
0.256350 + 0.966584i \(0.417480\pi\)
\(564\) 0 0
\(565\) 25.1216 1.05687
\(566\) 0 0
\(567\) 0.417424 0.0175302
\(568\) 0 0
\(569\) 30.1652 1.26459 0.632294 0.774728i \(-0.282113\pi\)
0.632294 + 0.774728i \(0.282113\pi\)
\(570\) 0 0
\(571\) −32.4955 −1.35989 −0.679946 0.733262i \(-0.737997\pi\)
−0.679946 + 0.733262i \(0.737997\pi\)
\(572\) 0 0
\(573\) −65.7042 −2.74483
\(574\) 0 0
\(575\) −18.3303 −0.764426
\(576\) 0 0
\(577\) −23.1216 −0.962564 −0.481282 0.876566i \(-0.659829\pi\)
−0.481282 + 0.876566i \(0.659829\pi\)
\(578\) 0 0
\(579\) 20.5826 0.855383
\(580\) 0 0
\(581\) 14.3739 0.596328
\(582\) 0 0
\(583\) −31.7477 −1.31486
\(584\) 0 0
\(585\) −14.3739 −0.594286
\(586\) 0 0
\(587\) 9.16515 0.378286 0.189143 0.981950i \(-0.439429\pi\)
0.189143 + 0.981950i \(0.439429\pi\)
\(588\) 0 0
\(589\) 7.37386 0.303835
\(590\) 0 0
\(591\) −6.62614 −0.272563
\(592\) 0 0
\(593\) −28.9129 −1.18731 −0.593655 0.804720i \(-0.702316\pi\)
−0.593655 + 0.804720i \(0.702316\pi\)
\(594\) 0 0
\(595\) −11.3739 −0.466283
\(596\) 0 0
\(597\) 66.2867 2.71294
\(598\) 0 0
\(599\) 20.3739 0.832453 0.416227 0.909261i \(-0.363352\pi\)
0.416227 + 0.909261i \(0.363352\pi\)
\(600\) 0 0
\(601\) 11.6261 0.474240 0.237120 0.971480i \(-0.423797\pi\)
0.237120 + 0.971480i \(0.423797\pi\)
\(602\) 0 0
\(603\) 44.9129 1.82899
\(604\) 0 0
\(605\) 10.1216 0.411501
\(606\) 0 0
\(607\) −26.4955 −1.07542 −0.537709 0.843131i \(-0.680710\pi\)
−0.537709 + 0.843131i \(0.680710\pi\)
\(608\) 0 0
\(609\) −10.5826 −0.428828
\(610\) 0 0
\(611\) 10.5826 0.428125
\(612\) 0 0
\(613\) 34.3739 1.38835 0.694174 0.719808i \(-0.255770\pi\)
0.694174 + 0.719808i \(0.255770\pi\)
\(614\) 0 0
\(615\) −31.7477 −1.28019
\(616\) 0 0
\(617\) −6.95644 −0.280056 −0.140028 0.990148i \(-0.544719\pi\)
−0.140028 + 0.990148i \(0.544719\pi\)
\(618\) 0 0
\(619\) −4.37386 −0.175800 −0.0879002 0.996129i \(-0.528016\pi\)
−0.0879002 + 0.996129i \(0.528016\pi\)
\(620\) 0 0
\(621\) −22.9129 −0.919462
\(622\) 0 0
\(623\) −7.58258 −0.303789
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) −10.5826 −0.422627
\(628\) 0 0
\(629\) 18.9564 0.755843
\(630\) 0 0
\(631\) 25.0000 0.995234 0.497617 0.867397i \(-0.334208\pi\)
0.497617 + 0.867397i \(0.334208\pi\)
\(632\) 0 0
\(633\) 68.0345 2.70413
\(634\) 0 0
\(635\) −20.2432 −0.803326
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) 58.2867 2.30579
\(640\) 0 0
\(641\) −20.2087 −0.798196 −0.399098 0.916908i \(-0.630677\pi\)
−0.399098 + 0.916908i \(0.630677\pi\)
\(642\) 0 0
\(643\) −23.0000 −0.907031 −0.453516 0.891248i \(-0.649830\pi\)
−0.453516 + 0.891248i \(0.649830\pi\)
\(644\) 0 0
\(645\) −16.7477 −0.659441
\(646\) 0 0
\(647\) 13.4174 0.527493 0.263747 0.964592i \(-0.415042\pi\)
0.263747 + 0.964592i \(0.415042\pi\)
\(648\) 0 0
\(649\) −46.1216 −1.81043
\(650\) 0 0
\(651\) 20.5826 0.806695
\(652\) 0 0
\(653\) 19.4174 0.759863 0.379931 0.925015i \(-0.375948\pi\)
0.379931 + 0.925015i \(0.375948\pi\)
\(654\) 0 0
\(655\) 66.3648 2.59309
\(656\) 0 0
\(657\) 78.4519 3.06070
\(658\) 0 0
\(659\) 36.9564 1.43962 0.719809 0.694172i \(-0.244229\pi\)
0.719809 + 0.694172i \(0.244229\pi\)
\(660\) 0 0
\(661\) −34.0000 −1.32245 −0.661223 0.750189i \(-0.729962\pi\)
−0.661223 + 0.750189i \(0.729962\pi\)
\(662\) 0 0
\(663\) −10.5826 −0.410993
\(664\) 0 0
\(665\) 3.00000 0.116335
\(666\) 0 0
\(667\) −17.3739 −0.672719
\(668\) 0 0
\(669\) 66.2867 2.56279
\(670\) 0 0
\(671\) −3.79129 −0.146361
\(672\) 0 0
\(673\) −17.1216 −0.659989 −0.329994 0.943983i \(-0.607047\pi\)
−0.329994 + 0.943983i \(0.607047\pi\)
\(674\) 0 0
\(675\) 20.0000 0.769800
\(676\) 0 0
\(677\) −21.9564 −0.843855 −0.421927 0.906630i \(-0.638646\pi\)
−0.421927 + 0.906630i \(0.638646\pi\)
\(678\) 0 0
\(679\) 7.00000 0.268635
\(680\) 0 0
\(681\) 10.5826 0.405525
\(682\) 0 0
\(683\) 36.1652 1.38382 0.691911 0.721983i \(-0.256769\pi\)
0.691911 + 0.721983i \(0.256769\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) 0 0
\(687\) −61.4083 −2.34287
\(688\) 0 0
\(689\) 8.37386 0.319019
\(690\) 0 0
\(691\) −48.7477 −1.85445 −0.927225 0.374504i \(-0.877813\pi\)
−0.927225 + 0.374504i \(0.877813\pi\)
\(692\) 0 0
\(693\) −18.1652 −0.690037
\(694\) 0 0
\(695\) −29.2432 −1.10926
\(696\) 0 0
\(697\) −14.3739 −0.544449
\(698\) 0 0
\(699\) −78.4955 −2.96897
\(700\) 0 0
\(701\) 42.3303 1.59879 0.799397 0.600804i \(-0.205153\pi\)
0.799397 + 0.600804i \(0.205153\pi\)
\(702\) 0 0
\(703\) −5.00000 −0.188579
\(704\) 0 0
\(705\) −88.6170 −3.33751
\(706\) 0 0
\(707\) 7.74773 0.291383
\(708\) 0 0
\(709\) 31.2432 1.17336 0.586681 0.809818i \(-0.300434\pi\)
0.586681 + 0.809818i \(0.300434\pi\)
\(710\) 0 0
\(711\) 47.9129 1.79687
\(712\) 0 0
\(713\) 33.7913 1.26549
\(714\) 0 0
\(715\) −11.3739 −0.425358
\(716\) 0 0
\(717\) 21.6261 0.807643
\(718\) 0 0
\(719\) 12.0000 0.447524 0.223762 0.974644i \(-0.428166\pi\)
0.223762 + 0.974644i \(0.428166\pi\)
\(720\) 0 0
\(721\) −13.0000 −0.484145
\(722\) 0 0
\(723\) −24.4174 −0.908094
\(724\) 0 0
\(725\) 15.1652 0.563220
\(726\) 0 0
\(727\) −11.0000 −0.407967 −0.203984 0.978974i \(-0.565389\pi\)
−0.203984 + 0.978974i \(0.565389\pi\)
\(728\) 0 0
\(729\) −43.8693 −1.62479
\(730\) 0 0
\(731\) −7.58258 −0.280452
\(732\) 0 0
\(733\) 41.4955 1.53267 0.766335 0.642441i \(-0.222078\pi\)
0.766335 + 0.642441i \(0.222078\pi\)
\(734\) 0 0
\(735\) 8.37386 0.308875
\(736\) 0 0
\(737\) 35.5390 1.30910
\(738\) 0 0
\(739\) 35.7477 1.31500 0.657501 0.753454i \(-0.271614\pi\)
0.657501 + 0.753454i \(0.271614\pi\)
\(740\) 0 0
\(741\) 2.79129 0.102541
\(742\) 0 0
\(743\) −21.3303 −0.782533 −0.391266 0.920277i \(-0.627963\pi\)
−0.391266 + 0.920277i \(0.627963\pi\)
\(744\) 0 0
\(745\) 9.00000 0.329734
\(746\) 0 0
\(747\) −68.8693 −2.51980
\(748\) 0 0
\(749\) 0.165151 0.00603450
\(750\) 0 0
\(751\) −11.6261 −0.424244 −0.212122 0.977243i \(-0.568037\pi\)
−0.212122 + 0.977243i \(0.568037\pi\)
\(752\) 0 0
\(753\) −27.3303 −0.995972
\(754\) 0 0
\(755\) −40.1216 −1.46017
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 0 0
\(759\) −48.4955 −1.76027
\(760\) 0 0
\(761\) −39.3303 −1.42572 −0.712861 0.701305i \(-0.752601\pi\)
−0.712861 + 0.701305i \(0.752601\pi\)
\(762\) 0 0
\(763\) 11.7477 0.425296
\(764\) 0 0
\(765\) 54.4955 1.97029
\(766\) 0 0
\(767\) 12.1652 0.439258
\(768\) 0 0
\(769\) −13.4955 −0.486659 −0.243329 0.969944i \(-0.578240\pi\)
−0.243329 + 0.969944i \(0.578240\pi\)
\(770\) 0 0
\(771\) 31.2867 1.12676
\(772\) 0 0
\(773\) 16.7477 0.602374 0.301187 0.953565i \(-0.402617\pi\)
0.301187 + 0.953565i \(0.402617\pi\)
\(774\) 0 0
\(775\) −29.4955 −1.05951
\(776\) 0 0
\(777\) −13.9564 −0.500684
\(778\) 0 0
\(779\) 3.79129 0.135837
\(780\) 0 0
\(781\) 46.1216 1.65036
\(782\) 0 0
\(783\) 18.9564 0.677448
\(784\) 0 0
\(785\) −60.3648 −2.15451
\(786\) 0 0
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) 0 0
\(789\) −18.9564 −0.674867
\(790\) 0 0
\(791\) −8.37386 −0.297740
\(792\) 0 0
\(793\) 1.00000 0.0355110
\(794\) 0 0
\(795\) −70.1216 −2.48696
\(796\) 0 0
\(797\) −32.8693 −1.16429 −0.582145 0.813085i \(-0.697787\pi\)
−0.582145 + 0.813085i \(0.697787\pi\)
\(798\) 0 0
\(799\) −40.1216 −1.41940
\(800\) 0 0
\(801\) 36.3303 1.28367
\(802\) 0 0
\(803\) 62.0780 2.19069
\(804\) 0 0
\(805\) 13.7477 0.484544
\(806\) 0 0
\(807\) −27.7913 −0.978300
\(808\) 0 0
\(809\) −22.9129 −0.805574 −0.402787 0.915294i \(-0.631958\pi\)
−0.402787 + 0.915294i \(0.631958\pi\)
\(810\) 0 0
\(811\) 31.4955 1.10595 0.552977 0.833196i \(-0.313492\pi\)
0.552977 + 0.833196i \(0.313492\pi\)
\(812\) 0 0
\(813\) 39.4174 1.38243
\(814\) 0 0
\(815\) −13.1216 −0.459629
\(816\) 0 0
\(817\) 2.00000 0.0699711
\(818\) 0 0
\(819\) 4.79129 0.167421
\(820\) 0 0
\(821\) 42.3303 1.47734 0.738669 0.674068i \(-0.235455\pi\)
0.738669 + 0.674068i \(0.235455\pi\)
\(822\) 0 0
\(823\) 54.2432 1.89080 0.945399 0.325915i \(-0.105672\pi\)
0.945399 + 0.325915i \(0.105672\pi\)
\(824\) 0 0
\(825\) 42.3303 1.47375
\(826\) 0 0
\(827\) −39.6606 −1.37913 −0.689567 0.724222i \(-0.742199\pi\)
−0.689567 + 0.724222i \(0.742199\pi\)
\(828\) 0 0
\(829\) −4.00000 −0.138926 −0.0694629 0.997585i \(-0.522129\pi\)
−0.0694629 + 0.997585i \(0.522129\pi\)
\(830\) 0 0
\(831\) 9.07803 0.314913
\(832\) 0 0
\(833\) 3.79129 0.131360
\(834\) 0 0
\(835\) −40.2523 −1.39299
\(836\) 0 0
\(837\) −36.8693 −1.27439
\(838\) 0 0
\(839\) −8.83485 −0.305013 −0.152506 0.988302i \(-0.548734\pi\)
−0.152506 + 0.988302i \(0.548734\pi\)
\(840\) 0 0
\(841\) −14.6261 −0.504350
\(842\) 0 0
\(843\) 54.1996 1.86673
\(844\) 0 0
\(845\) −36.0000 −1.23844
\(846\) 0 0
\(847\) −3.37386 −0.115927
\(848\) 0 0
\(849\) 17.7913 0.610595
\(850\) 0 0
\(851\) −22.9129 −0.785443
\(852\) 0 0
\(853\) 9.12159 0.312317 0.156159 0.987732i \(-0.450089\pi\)
0.156159 + 0.987732i \(0.450089\pi\)
\(854\) 0 0
\(855\) −14.3739 −0.491576
\(856\) 0 0
\(857\) −11.7042 −0.399807 −0.199903 0.979816i \(-0.564063\pi\)
−0.199903 + 0.979816i \(0.564063\pi\)
\(858\) 0 0
\(859\) −7.37386 −0.251593 −0.125796 0.992056i \(-0.540149\pi\)
−0.125796 + 0.992056i \(0.540149\pi\)
\(860\) 0 0
\(861\) 10.5826 0.360653
\(862\) 0 0
\(863\) 38.7042 1.31751 0.658753 0.752360i \(-0.271084\pi\)
0.658753 + 0.752360i \(0.271084\pi\)
\(864\) 0 0
\(865\) 68.2432 2.32034
\(866\) 0 0
\(867\) −7.33030 −0.248950
\(868\) 0 0
\(869\) 37.9129 1.28611
\(870\) 0 0
\(871\) −9.37386 −0.317621
\(872\) 0 0
\(873\) −33.5390 −1.13512
\(874\) 0 0
\(875\) 3.00000 0.101419
\(876\) 0 0
\(877\) −38.7477 −1.30842 −0.654209 0.756314i \(-0.726998\pi\)
−0.654209 + 0.756314i \(0.726998\pi\)
\(878\) 0 0
\(879\) 55.1216 1.85921
\(880\) 0 0
\(881\) 2.37386 0.0799775 0.0399887 0.999200i \(-0.487268\pi\)
0.0399887 + 0.999200i \(0.487268\pi\)
\(882\) 0 0
\(883\) −24.2523 −0.816154 −0.408077 0.912948i \(-0.633801\pi\)
−0.408077 + 0.912948i \(0.633801\pi\)
\(884\) 0 0
\(885\) −101.869 −3.42430
\(886\) 0 0
\(887\) −4.58258 −0.153868 −0.0769339 0.997036i \(-0.524513\pi\)
−0.0769339 + 0.997036i \(0.524513\pi\)
\(888\) 0 0
\(889\) 6.74773 0.226312
\(890\) 0 0
\(891\) −1.58258 −0.0530183
\(892\) 0 0
\(893\) 10.5826 0.354132
\(894\) 0 0
\(895\) 47.3739 1.58353
\(896\) 0 0
\(897\) 12.7913 0.427089
\(898\) 0 0
\(899\) −27.9564 −0.932399
\(900\) 0 0
\(901\) −31.7477 −1.05767
\(902\) 0 0
\(903\) 5.58258 0.185776
\(904\) 0 0
\(905\) −28.1216 −0.934793
\(906\) 0 0
\(907\) −31.2432 −1.03741 −0.518706 0.854952i \(-0.673586\pi\)
−0.518706 + 0.854952i \(0.673586\pi\)
\(908\) 0 0
\(909\) −37.1216 −1.23125
\(910\) 0 0
\(911\) −3.49545 −0.115810 −0.0579048 0.998322i \(-0.518442\pi\)
−0.0579048 + 0.998322i \(0.518442\pi\)
\(912\) 0 0
\(913\) −54.4955 −1.80354
\(914\) 0 0
\(915\) −8.37386 −0.276831
\(916\) 0 0
\(917\) −22.1216 −0.730519
\(918\) 0 0
\(919\) 34.4955 1.13790 0.568950 0.822372i \(-0.307350\pi\)
0.568950 + 0.822372i \(0.307350\pi\)
\(920\) 0 0
\(921\) 76.4083 2.51774
\(922\) 0 0
\(923\) −12.1652 −0.400421
\(924\) 0 0
\(925\) 20.0000 0.657596
\(926\) 0 0
\(927\) 62.2867 2.04577
\(928\) 0 0
\(929\) 47.8693 1.57054 0.785271 0.619153i \(-0.212524\pi\)
0.785271 + 0.619153i \(0.212524\pi\)
\(930\) 0 0
\(931\) −1.00000 −0.0327737
\(932\) 0 0
\(933\) −73.1561 −2.39502
\(934\) 0 0
\(935\) 43.1216 1.41023
\(936\) 0 0
\(937\) −15.3739 −0.502242 −0.251121 0.967956i \(-0.580799\pi\)
−0.251121 + 0.967956i \(0.580799\pi\)
\(938\) 0 0
\(939\) 35.5826 1.16119
\(940\) 0 0
\(941\) 27.4955 0.896326 0.448163 0.893952i \(-0.352078\pi\)
0.448163 + 0.893952i \(0.352078\pi\)
\(942\) 0 0
\(943\) 17.3739 0.565771
\(944\) 0 0
\(945\) −15.0000 −0.487950
\(946\) 0 0
\(947\) 52.6170 1.70982 0.854912 0.518773i \(-0.173611\pi\)
0.854912 + 0.518773i \(0.173611\pi\)
\(948\) 0 0
\(949\) −16.3739 −0.531518
\(950\) 0 0
\(951\) 41.4083 1.34276
\(952\) 0 0
\(953\) 21.4610 0.695189 0.347595 0.937645i \(-0.386999\pi\)
0.347595 + 0.937645i \(0.386999\pi\)
\(954\) 0 0
\(955\) −70.6170 −2.28511
\(956\) 0 0
\(957\) 40.1216 1.29695
\(958\) 0 0
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) 23.3739 0.753996
\(962\) 0 0
\(963\) −0.791288 −0.0254989
\(964\) 0 0
\(965\) 22.1216 0.712119
\(966\) 0 0
\(967\) −12.1216 −0.389804 −0.194902 0.980823i \(-0.562439\pi\)
−0.194902 + 0.980823i \(0.562439\pi\)
\(968\) 0 0
\(969\) −10.5826 −0.339961
\(970\) 0 0
\(971\) −15.0000 −0.481373 −0.240686 0.970603i \(-0.577373\pi\)
−0.240686 + 0.970603i \(0.577373\pi\)
\(972\) 0 0
\(973\) 9.74773 0.312498
\(974\) 0 0
\(975\) −11.1652 −0.357571
\(976\) 0 0
\(977\) −34.4174 −1.10111 −0.550555 0.834799i \(-0.685584\pi\)
−0.550555 + 0.834799i \(0.685584\pi\)
\(978\) 0 0
\(979\) 28.7477 0.918781
\(980\) 0 0
\(981\) −56.2867 −1.79710
\(982\) 0 0
\(983\) −8.66970 −0.276520 −0.138260 0.990396i \(-0.544151\pi\)
−0.138260 + 0.990396i \(0.544151\pi\)
\(984\) 0 0
\(985\) −7.12159 −0.226913
\(986\) 0 0
\(987\) 29.5390 0.940237
\(988\) 0 0
\(989\) 9.16515 0.291435
\(990\) 0 0
\(991\) 8.74773 0.277881 0.138940 0.990301i \(-0.455630\pi\)
0.138940 + 0.990301i \(0.455630\pi\)
\(992\) 0 0
\(993\) −79.1996 −2.51332
\(994\) 0 0
\(995\) 71.2432 2.25856
\(996\) 0 0
\(997\) −2.87841 −0.0911601 −0.0455801 0.998961i \(-0.514514\pi\)
−0.0455801 + 0.998961i \(0.514514\pi\)
\(998\) 0 0
\(999\) 25.0000 0.790965
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2128.2.a.k.1.2 2
4.3 odd 2 532.2.a.c.1.1 2
8.3 odd 2 8512.2.a.t.1.2 2
8.5 even 2 8512.2.a.m.1.1 2
12.11 even 2 4788.2.a.g.1.2 2
28.27 even 2 3724.2.a.e.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
532.2.a.c.1.1 2 4.3 odd 2
2128.2.a.k.1.2 2 1.1 even 1 trivial
3724.2.a.e.1.2 2 28.27 even 2
4788.2.a.g.1.2 2 12.11 even 2
8512.2.a.m.1.1 2 8.5 even 2
8512.2.a.t.1.2 2 8.3 odd 2